Sparse representation (SR) methods in quaternion space have been attracting increasing interests recently. However, most existing quaternion SR methods adopt the quaternion ℓ norm, which penalizes all the entries of the quaternion sparse vector equally and ignores the differences and significance of different entries. Ideally, the entries with large magnitude should be less penalized while those with small magnitude (such as zero entries) should be more penalized. Therefore, we propose an Adaptive Weighted Quaternion Sparse Representation (AWQSR) method in this paper, which can learn weights for distinct entries of the quaternion sparse entries in an adaptive manner. Due to the noncommutativity of quaternion multiplication, it is difficult to tackle the resulting optimization problem of AWQSR. For this reason, we devise an effective iteratively reweighted optimization algorithm based on quaternion operators. To further improve the classification performance, we also develop a Supervised AWQSR based Classification (SAWQSRC) method by leveraging the label information of training samples to learn discriminative weights. Theoretical analysis of SAWQSRC has also been established to show that SAWQSRC succeeds in classification under appropriate conditions. The experiments on simulated data and real data prove the validity of the proposed methods for quaternion signal recovery and classification.